Chapter 6.3, Problem 10E

(a)

To determine

To calculate: The reference angle of $2.3\pi$ .

Answer to Problem 10E

The reference angle of $2.3\pi$ is $\frac{2\pi }{7}$ .

Explanation of Solution

Given information:

The angle $2.3\pi$ .

Formula used:

The reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

Calculation:

Consider the provided angle $2.3\pi$ .

Denote $\theta =2.3\pi$ .

Convert into degrees,

  $\begin{array}{c}\theta =2.3\pi \\ \theta =2.3\pi \cdot \frac{180}{\pi }\\ \theta ={414}^{\circ }\end{array}$

Recall that the reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

From the figure provided below,

  

The provided angle has terminal side in Quadrant $\text{I}$ .

The reference angle is,

  $\begin{array}{c}\overline{\theta }={414}^{\circ }-{360}^{\circ }\\ ={54}^{\circ }\\ =0.3\pi \end{array}$

Thus, the reference angle of $2.3\pi$ is $0.3\pi$ .

(b)

To determine

To calculate: The reference angle of $2.3$ .

Answer to Problem 10E

The reference angle of $2.3$ is $1.4$ .

Explanation of Solution

Given information:

The angle $2.3$ .

Formula used:

The reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

Calculation:

Consider the provided angle $2.3$ .

Denote $\theta =2.3$ .

First convert $2.3\text{ radians}$ into degrees,

  $\begin{array}{l}\theta =2.3\\ \theta =2.3\cdot \frac{180}{\pi }\\ \theta \approx {132}^{\circ }\end{array}$

Recall that the reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

From the figure provided below,

  

The provided angle has terminal side in Quadrant $\text{II}$ .

The reference angle is,

  $\begin{array}{c}\overline{\theta }={180}^{\circ }-{132}^{\circ }\\ ={48}^{\circ }\\ =0.84\end{array}$

Thus, the reference angle of $2.3$ is $0.84$ .

(c)

To determine

To calculate: The reference angle of $-10\pi$ .

Answer to Problem 10E

The reference angle of $-10\pi$ is $0^{\circ }$ .

Explanation of Solution

Given information:

The angle $-10\pi$ .

Formula used:

The reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

Calculation:

Consider the provided angle $-10\pi$ .

Denote $\theta =-10\pi$ .

First convert $-10\pi$ into degrees,

  $\begin{array}{l}\theta =-10\pi \\ \theta =-10\pi \cdot \frac{180}{\pi }\\ \theta =-{1800}^{\circ }\end{array}$

Recall that the reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

From the figure provided below,

  

The provided angle has terminal side on positive $x-axis$ .

The reference angle is,

  $\begin{array}{c}\overline{\theta }=5\left({360}^{\circ }\right)-{1800}^{\circ }\\ =0^{\circ }\end{array}$

Thus, the reference angle of $-10\pi$ is $0^{\circ }$ .